# Josherich's Blog

## Euler's formula as power series

21 Mar 2019

$e^{i\theta}$ is the points on unit circle, at angle $\theta$, here is a visualization of how such circle is derived from the sum of power series:

Assume we have a function with the property of its derivative is equal to itself, just as $\frac{d}{dx}e^x = e^x$, such function should have the form:

Expand to complex number $i\theta$:

Draw these complex numbers as vectors on complex plane, we have a end point for each $\theta$, and a unit circle with all $\theta$s

move mouse horizontally to draw the circle
function drawETheta(theta, iter, scale) {
var itheta = new Complex(0, theta);
var res = new Complex(1, 0).add(itheta);

var path = new Path();
path.strokeColor = 'black';